An undetermined system may have multiple or infinite solutions, in opposition to a determined system with a single unique solution. Such systems may find use in the emerging concept of applying System-on-Chip (SoC) to the case of Radio-on-Chip (software defined radio) in wireless base stations. Such systems may apply adaptive equalizers, linearizers or identifiers in either the transmitter or receiver or both.
These systems can be decomposed into a plant and model. The plant represents the physical system to be corrected (such as, but not limited to, a nonlinear transmitter) or identified and the model represents the artificial structure to be adapted to correct (through inversion) or mimic (through modeling) the plant, depending on the system architecture. The models are ideally trained (adapted) in a test or characterization mode, whereby the system is taken out of service periodically and a known test waveform applied to the system that is of similar frequency bandwidth as the plant bandwidth. However, the conflicting requirements to minimize system down-time while providing a suitable training frequency to maintain feature performance over time, precludes a characterization mode. There is a need to be able to provide adaptation with the transmission signal.
As modern radio products must support a variety of signal bandwidths, including narrow bandwidth signals, there exists the possibility for the plant bandwidth to be significantly larger than the signal bandwidth. In this case, there is insufficient information to accurately solve the system of equations characterizing the plant and the associated model—there are in effect more unknowns than equations. This scenario is described in mathematics as an under-determined system. The severity of under-determinedness increases with model complexity (model dimensionality and span—more unknowns) and excitation signal correlation (narrow bandwidth—less information).
A model solution can be found through block-based processing where data is collected in blocks, processed directly to solve for the model parameters (solution) which are then applied to the model. Other attempts to provide a model solution involve gradient methods where an error signal is processed sample-by-sample, with each outcome driving directly the model parameters towards a minimized error and ultimately the solution. When applied to solve an under-determined system, both methods can be impaired and may be sensitive to bandwidth and type of model used. While both methods of adaptation are valid, they may also lack efficiency and robustness.